`lim_(x->oo)(e^(11x)-7x)^(1/(3x))`

Options

A. `(11)/(3)`
B. `(3)/(11)`
C. `e^((3)/(11))`
D. `e ^((11)/(3))`

Answer

To solve the limit \( \lim_{x \to \infty} (e^{11x} - 7x)^{\frac{1}{3x}} \), we will follow these steps:

### Step 1: Identify the limit
We start by letting:
\[ L = \lim_{x \to \infty} (e^{11x} - 7x)^{\frac{1}{3x}} \]

### Step 2: Take the natural logarithm
Taking the natural logarithm of both sides:
\[ \ln L = \lim_{x \to \infty} \frac{1}{3x} \ln(e^{11x} - 7x) \]

### Step 3: Analyze the logarithm
As \( x \to \infty \), \( e^{11x} \) grows much faster than \( 7x \). Thus, we can approximate:
\[ \ln(e^{11x} - 7x) \approx \ln(e^{11x}) = 11x \]
So we rewrite the limit:
\[ \ln L = \lim_{x \to \infty} \frac{1}{3x} \cdot 11x = \lim_{x \to \infty} \frac{11}{3} = \frac{11}{3} \]

### Step 4: Exponentiate to find \( L \)
Now, we exponentiate both sides to solve for \( L \):
\[ L = e^{\frac{11}{3}} \]

### Conclusion
Thus, the limit is:
\[ \lim_{x \to \infty} (e^{11x} - 7x)^{\frac{1}{3x}} = e^{\frac{11}{3}} \]

### Final Answer
The correct answer is \( e^{\frac{11}{3}} \).

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